A number of common objects incorporate optical security devices conferring authenticity to the object. For example, banknotes contain certain regions which change colors when the position from which they are observed or seen is changed. These devices are generally flat structures producing effects in the incident light and allow identifying the genuineness of an object to the naked eye. However, they can be counterfeited by using similar structures or other less sophisticated structures producing a similar response such that the counterfeiting cannot be identified as such to the naked eye. Therefore, a primary objective of optical security devices is for them to produce an optical response preventing the counterfeiting thereof. In other words, a structure must be created the optical response (or signature) of which cannot be synthesized by other means. There are a number of techniques for producing optical security marks of this type, such as the one disclosed in United States patent application US-A-20030058491, for example.
On the other hand, metamaterials have become one of the most relevant scientific topics today. To better understand the basic electromagnetic properties of metamaterials, it is first necessary to consider how natural media respond to electromagnetic radiation. When the incident electromagnetic radiation on a natural medium (for example, quartz or water) has a wavelength that is much greater than the size of the atoms/molecules (of the order of several Armstrong) that form it, the medium has an effective response that is characterized by two fundamental physical magnitudes: electric permittivity ε=εrε0 (εr is the relative permittivity and ε0 permittivity of free space), which models the response of the medium to the electric field; and magnetic permeability μ=μrμ0 (μr is the relative permeability and μ0 the permeability of free space), which models the response of the medium to the magnetic field. Using these magnitudes, the unique properties of the medium found in Maxwell's equations, which can be complex numbers, the refractive index of the medium is defined as n=(εrμr)1/2 and the impedance of the medium is defined as η=(μ/ε)1/2. In the region of dielectric media transparent to optical frequencies (above 100 THz), εr is a positive real number greater than 1 whereas μr=1 because natural media do not show magnetic activity at optical frequencies. In the case of metals below the plasma frequency (generally in the visible or ultraviolet range), εr is a negative number the modulus of which increases inversely to the frequency, and εr=1.
By using such natural materials with these electromagnetic characteristics it is possible to obtain structures with certain properties or functionalities, such as waveguides, for example, for carrying light between two points in a confined manner. However, there is a limit in terms of the values of the parameters n and η that can be obtained because a natural material cannot be altered in terms of its physical nature in order to vary its electromagnetic properties. Therefore, in terms of the design of the structures, it is limited to the values of n and ri of the materials that can be found in nature. For example, there are no natural materials with magnetic activity at optical frequencies (μr≠1). In this sense, the design capacity is very limited.
In contrast, a metamaterial is an artificial medium formed by meta-atoms of a much smaller size (at least in the electromagnetic field propagation direction) than the wavelength λ of the incident radiation and the electromagnetic response of which depends not only on the electromagnetic properties of the media forming them but on how the aforementioned meta-atoms are structured. In general, said meta-atoms make up a disordered or periodic structure with a certain period ai (i=x, y, z) in each of the directions of the space, x, y, z. The size of the meta-atoms is much greater than that of a natural atom or molecule, the periods a, also being much greater than the interatomic distance in natural substances. A metamaterial can theoretically be designed such that it has any imaginable value of the effective electric permittivity εr and of the magnetic permeability μr, from infinite to zero, and both positive and negative values. Accordingly, any imaginable value of the parameters n and η can also be obtained. In other words, metamaterials allow synthesizing “custom-made” electromagnetic media.
The origin of metamaterials can be found in a theoretical article published by Russian physicist V. Veselago from 40 years ago [V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of permittivity and permeability,” Soy. Phys. Usp. 10, 509 (1968)]. In this document, Veselago studied the inverse properties of ideal electromagnetic media (in the sense that they are homogenous, isotropic and loss-free) with simultaneously negative electric permittivity and magnetic permeability, and it was concluded that if both properties had a negative sign, the refractive index n must as well. Given that at that time there were no natural or artificial materials with these properties (the typical electromagnetic properties of metals and dielectric materials, the most typical substances found in nature in terms of electromagnetic behavior, have already been discussed above), this work remained forgotten for over 30 years until it was salvaged by English scientist Sir John Pendry.
The idea was to develop artificial materials (hence the name metamaterials) the electric and magnetic responses of which could be designed to produce any imaginable value. First, Pendry demonstrated that a three-dimensional lattice of metal wires has a diluted plasmon response, such that the plasma frequency (frequency from which the medium is transparent) depends not on the metal with which the lattice is made but on the periodicity thereof an on the radius of the wires [J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely Low Frequency Plasmons in Metallic Mesostructures,” Phys. Rev. Lett. 76, 4773 (1996)]. Shortly thereafter, Pendry proposed that two concentrically split metal rings have a resonant behavior at a certain frequency in which the effective magnetic permeability experiences a very abrupt change, even reaching negative values [J. B. Pendry, A. J. Holden, D. J. Robins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Technol. 47, 2075 (1999)]. By mixing both structures, the first experimental demonstration of the negative refraction phenomenon using a metamaterial with εr, μr<0 simultaneously was performed in 2001 at microwave frequencies [R. A. Shelby, D. R. Smith, S. Schultz, “Experimental Verification of a Negative Index of Refraction,” Science 292, 77-79 (2002)]. In general, it can be said that the first custom-designed metamaterial having a refractive index n and which was impossible to obtain with a natural media was made.
In the microwave regimen, the periods a, are of the order of centimeters or millimeters. Through the scaling properties of Maxwell's equations, it can be considered that by reducing those periods a, to orders of micrometers or hundreds of nanometers, it is possible to obtain metamaterials with a “custom-made” response at optical frequencies (visible, infrared). This is true only in part because the metals used to build the metamaterial in the aforementioned paper by Shelby et al. behave like perfect conductors in microwaves whereas at optical frequencies they are characterized by the existence of surface plasmons which complicate making metamaterials at such high frequencies. Furthermore, for other reasons there is a maximum frequency at which a magnetic response can be obtained with the ring resonators proposed by Pendry due to the saturation of the magnetic response [J. Zhou, T H. Koschny, M. Kafesaki, E. N. Economou, J. B. Pendry and C. M. Soukoulis, “Saturation of the negative magnetic response of split-ring resonators at optical frequencies,” Phys. Rev. Lett. 95, 223902 (2005)]. In addition to the foregoing, the difficulty in manufacturing meta-atoms having such small sizes requiring very complex and advanced nanomanufacturing processes must be taken into account. Accordingly, making three-dimensional metamaterials with a magnetic response at optical frequencies (mainly near-infrared and visible) is still a challenge. It must be pointed out that metamaterials are the only way to produce magnetic activity (μr≠1) at optical frequencies at which all natural materials are inert to the magnetic field.
However, different experiments have demonstrated the possibility of making planar metamaterials, i.e., two-dimensional materials having one or several layers with a magnetic response at optical frequencies [T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz Magnetic Response from Artificial Materials,” Science 303, 1494-1496 (2004); V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photon. 1, 41-48 (2007); S. Zhang, W. Fan, A. Frauenglass, B. Minhas, K. J. Malloy and S. R. J. Brueck, “Demonstration of Mid-Infrared Resonant Magnetic Nanostructures Exhibiting a Negative Permeability,” Phys. Rev. Lett. 94, 037402 (2005)]. Despite the slenderness in terms of wavelengths of said layers of metamaterial, extraction algorithms of the parameters εr and μr, determined univocally from the transmission and reflection measurements upon illuminating the aforementioned layer with both normal and oblique incidence can be used. Said algorithms are described in depth in several articles: [D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002); C. Menzel, C. Rockstuhl, T. Paul, F. Lederer, and T. Pertsch “Retrieving effective parameters for metamaterials at oblique incidence”, Phys. Rev. B 77, 195328 (2008)]. By using this method it has been possible to determine that said layers have an exclusive behavior of metamaterials, including a relative permeability different from 1 at optical frequencies.
In a recent design the feasibility of obtaining a simultaneous electric and magnetic response in the visible spectrum has been theoretically demonstrated using a silver metamaterial without needing to simultaneously use split-ring resonators and metal strips [C. Garcia-Meca, R. Orturio, R. Salvador, A. Martinez, and J. Marti, “Low-loss single-layer metamaterial with negative index of refraction at visible wavelengths,” Opt. Express 15, 9320-9325 (2007)].
Document WO-A-2008/110775 discloses security marks based on different structures of metamaterials and essentially corresponding to two types of configurations: one for refraction and the other for diffraction of radiations in the terahertz range (wavelength of 3 mm to 15 μm) or infrared range (wavelength greater than 750 nm). Even though the metamaterials present in these structures provide responses in diffraction and/or refraction different from those of natural media in said diffraction and refraction configurations, they have the drawback that those responses are imitable using materials of another type such as, for example, photonic crystals (periodic dielectric structures).
Document WO-A-2006023195 discloses metamaterials for use in optical devices such as lenses formed from a plurality of unit cells at least a portion of which has an electromagnetic permeability different from others and arranged such that the material has a gradient index such that a continuous variation of the permeability takes place, which does not allow forming an effective matrix security code which would be required of a discrete variation of the permeability.